The Cosmic Microwave Background Radiation (CMBR), which is a
direct relic of the early Universe, currently provides the
deepest probe of cosmological structures and imposes severe
constraints on the various proposed matter evolution scenarios
and cosmological parameters. Although the CMBR is a unique and
deep probe of both the thermal history of the early Universe and
the primordial perturbations in the matter distribution, the
associated anisotropies are not exclusively primordial in nature.
Important modifications to the CMBR spectrum can arise from large
scale coherent structures, even well after the photons decouple
from the matter at redshift
, due to the gravitational redshifting of the photons through
the Sachs-Wolfe effect arising from potential gradients [50,
5]

where the integral is evaluated from the emission (e) to reception (r) points along the spatial photon paths
,
is the gravitational potential,
defines the temperature fluctuations, and
a
(t) is the cosmological scale factor in the standard FLRW metric.
Also, if the intergalactic medium (IGM) reionizes sometime after
the decoupling, say from an early generation of stars, the
increased rate of Thomson scattering off the free electrons will
erase sub-horizon scale temperature anisotropies, while creating
secondary Doppler shift anisotropies. To make meaningful
comparisons between numerical models and observed data, all of
these effects (and others, see for example §
4.1.3) must be incorporated self-consistently into the numerical
models and to high accuracy in order to resolve the weak
signals.

Many computational analyses based on linear perturbation theory
have been carried out to estimate the temperature anisotropies in
the sky (for example see [41] and the references cited in [32]). Although such linearized approaches yield reasonable results,
they are not well-suited to discussing the expected imaging of
the developing
nonlinear
structures in the microwave background. An alternative
ray-tracing approach has been developed by Anninos et al. [5] to introduce and propagate individual photons through the
evolving nonlinear matter structures. They solve the geodesic
equations of motion and subject the photons to Thomson scattering
in a probabilistic way and at a rate determined by the local
density of free electrons in the model. Since the temperature
fluctuations remain small, the equations of motion for the
photons are treated as in the linearized limit, and the
anisotropies are computed according to

where

and the photon wave vector
and matter rest frame four-velocity
are evaluated at the emission (e) and reception (r) points.
Applying their procedure to a Hot Dark Matter (HDM) model of
structure formation, Anninos et al. [5] find the parameters for this model are severely constrained by
COBE data such that
, where
and
h
are the density and Hubble parameters.

In models where the IGM does not reionize, the probability of
scattering after the photon-matter decoupling epoch is low, and
the Sachs-Wolfe effect dominates the anisotropies at angular
scales larger than a few degrees. However if reionization occurs,
the scattering probability increases substantially and the matter
structures, which develop large bulk motions relative to the
comoving background, induce Doppler shifts on the scattered CMBR
photons and leave an imprint of the surface of last scattering.
The induced fluctuations on subhorizon scales in reionization
scenarios can be a significant fraction of the primordial
anisotropies, as observed by Tuluie et al. [56]. They considered two possible scenarios of reionization: A
model that suffers early and gradual (EG) reionization of the IGM
as caused by the photoionizing UV radiation emitted by decaying
neutrinos, and the late and sudden (LS) scenario as might be
applicable to the case of an early generation of star formation
activity at high redshifts. Considering the HDM model with
and
h
=0.55, which produces CMBR anisotropies above current COBE limits
when no reionization is included (see §
4.1.1), they find that the EG scenario effectively reduces the
anisotropies to the levels observed by COBE and generates smaller
Doppler shift anisotropies than the LS model, as demonstrated in
Figure
3
.

Additional sources of CMBR anisotropy can arise from the
interactions of photons with dynamically evolving matter
structures and nonstatic gravitational potentials. Tuluie et al.
[55] considered the impact of nonlinear matter condensations on the
CMBR in
Cold Dark Matter (CDM) models, focusing on the relative
importance of secondary temperature anisotropies due to three
different effects: 1) time-dependent variations in the
gravitational potential of nonlinear structures as a result of
collapse or expansion; 2) proper motion of nonlinear structures
such as clusters and superclusters across the sky; and 3) the
decaying gravitational potential effect from the evolution of
perturbations in open models. They applied the ray-tracing
procedure of [5] to explore the relative importance of these secondary
anisotropies as a function of the density parameter
and the scale of matter distributions. They find that the
secondary temperature anisotropies are dominated by the decaying
potential effect at large scales, but that all three sources of
anisotropy can produce signatures of order
as shown in Figure
4
.

Figure 4:
The top two images represent the proper motion and Rees-Sciama
effects in the CMBR for a critically closed CDM model (upper
left), together with the corresponding column density of voids
and clusters over the same region (upper right). The bottom two
images show the secondary anisotropies dominated here by the
decaying potential effect in an open cosmological model (bottom
left), together with the corresponding gravitational potential
over the same region (bottom right). The rms fluctuations in both
cases are on the order of
, though the open model carries a somewhat larger signature.

In addition to the effects discussed in the previous
paragraphs, many other sources of secondary anisotropies (such as
gravitational lensing, the Vishniac effect accounting for matter
velocities and flows into local potential wells, and the
Sunyaev-Zel'dovich distortions from the Compton scattering of CMB
photons in the hot cluster medium) can also be significant. See
reference [32] for a more complete list and thorough discussion of the
different sources of CMBR anisotropies.